Isometric immersion
= Isometric immersion
{title2=$f^*g_N=g_M$}
An isometric immersion is an <immersion> $f:M\to N$ between <Riemannian manifolds> whose derivative preserves the <inner products> of <tangent vectors>; equivalently, $f^*g_N=g_M$. Its derivative is injective and the target may have larger dimension. A <local isometry> additionally requires a <local diffeomorphism>, so its tangent maps are <linear isomorphisms>. For example, a <sphere> with its induced <Riemannian metric> is isometrically immersed in <Euclidean space>, although its positive intrinsic <sectional curvature> differs from that of the ambient space.